TY - JOUR

T1 - Signal recovery from a few linear measurements of its high-order spectra

AU - Bendory, Tamir

AU - Edidin, Dan

AU - Kreymer, Shay

N1 - Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2022/1

Y1 - 2022/1

N2 - The q-th order spectrum is a polynomial of degree q in the entries of a signal x∈CN, which is invariant under circular shifts of the signal. For q≥3, this polynomial determines the signal uniquely, up to a circular shift, and is called a high-order spectrum. The high-order spectra, and in particular the bispectrum (q=3) and the trispectrum (q=4), play a prominent role in various statistical signal processing and imaging applications, such as phase retrieval and single-particle reconstruction. However, the dimension of the q-th order spectrum is Nq−1, far exceeding the dimension of x, leading to increased computational load and storage requirements. In this work, we show that it is unnecessary to store and process the full high-order spectra: a signal can be uniquely characterized up to symmetries, from only N+1 linear measurements of its high-order spectra. The proof relies on tools from algebraic geometry and is corroborated by numerical experiments.

AB - The q-th order spectrum is a polynomial of degree q in the entries of a signal x∈CN, which is invariant under circular shifts of the signal. For q≥3, this polynomial determines the signal uniquely, up to a circular shift, and is called a high-order spectrum. The high-order spectra, and in particular the bispectrum (q=3) and the trispectrum (q=4), play a prominent role in various statistical signal processing and imaging applications, such as phase retrieval and single-particle reconstruction. However, the dimension of the q-th order spectrum is Nq−1, far exceeding the dimension of x, leading to increased computational load and storage requirements. In this work, we show that it is unnecessary to store and process the full high-order spectra: a signal can be uniquely characterized up to symmetries, from only N+1 linear measurements of its high-order spectra. The proof relies on tools from algebraic geometry and is corroborated by numerical experiments.

UR - http://www.scopus.com/inward/record.url?scp=85116854333&partnerID=8YFLogxK

U2 - 10.1016/j.acha.2021.10.003

DO - 10.1016/j.acha.2021.10.003

M3 - מזכר

AN - SCOPUS:85116854333

VL - 56

SP - 391

EP - 401

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

SN - 1063-5203

ER -